MIMO Basics

M8-L3 spent an entire lesson establishing that multipath is the enemy. Copies of the same signal arrive by different routes, add with random phases, and the sum collapses — 9.5% of the time a Rayleigh channel sits more than 10 dB below its own mean, which is where the fade margin of M9-L4 came from. This lesson performs the single most surprising inversion in the course. Multipath is not the enemy. It is the resource. Rich scattering is precisely what creates several independent paths between one end of a link and the other, and a radio with several antennas at each end can use those independent paths as separate channels in the same hertz at the same time. The very environment that made a single-antenna link unreliable is what makes a multi-antenna link fast. A clean line of sight, the ideal of Module 8, turns out to be the worst possible case for this trick.

The Inversion, Stated Precisely

MIMO — multiple input, multiple output — means N_t transmit antennas and N_r receive antennas operating as one system, on one carrier, in one channel. With a single antenna at each end there is one complex number describing the channel: an amplitude and a phase, fading as M8-L3 described. With several antennas there is one such number for every transmit–receive pair, so the channel becomes a matrix:

That last clause carries the whole lesson. Independence of the equations is not free: it is a property of the propagation environment. If every transmit antenna reaches every receive antenna by essentially the same route — one clear line of sight — then every row of the matrix is a scaled copy of every other row, the equations are redundant, and there is only ever one channel there no matter how many antennas you bolt on. Independence requires scattering. This is why the same phrase — rich scattering — that meant “bad news” in Module 8 means “capacity” here.

Three Gains, and You Cannot Have All Three

A set of antennas can be spent in three different ways. They are genuinely different purchases, they are bought with the same hardware, and they compete for it — which is why a datasheet that claims all three at once is describing three configurations rather than one. M7-L4 already drew the first line of this distinction and deferred the rest to this lesson.

Spatial multiplexing buys rate

Send different data on each transmit antenna. The receiver, knowing the matrix, unmixes the streams. This is the gain that multiplies throughput, it needs the channels to be decorrelated, and it is the only one of the three that can push a link above the single-channel Shannon bound of M6-L1 — which is exactly what M6-L3 promised to explain here. Shannon is not broken; there are simply several channels, and each obeys it separately.

Diversity buys reliability

Send the same data over several paths, so that a fade has to hit all of them simultaneously in order to matter. This gain does not raise the peak rate at all — it removes the deep tail of the fading distribution, which is often worth far more, because M9-L4’s fade margin is a direct tax on range. Diversity also needs decorrelated channels: two antennas that fade together provide no diversity whatsoever.

Array gain buys SNR

Weight the antennas so their contributions add coherently in one direction — beamforming, taught in full in M7-L4. This raises the signal-to-noise ratio, and unlike the other two it works best when the channels are correlated, because a beam is precisely a coherent sum. That is the tension in one sentence: beamforming wants the antennas to see the same channel, and spatial multiplexing wants them to see different ones. M7-L4 stated this distinction; it is worth being blunt about the consequence, which is that the same eight antennas cannot simultaneously be a maximally focused beam and eight independent streams.

The formal statement of the conflict is the multiplexing–diversity tradeoff: for a given number of antennas there is a frontier of achievable (rate, reliability) pairs, and moving along it trades one for the other. A 2×2 system can carry two streams with no diversity per stream, or one stream with diversity order 4, or something in between — but not two streams each with diversity order 4. Real systems move along this frontier dynamically: when the reported channel quality is good they add streams, and when it degrades they collapse to one stream and spend the antennas on robustness instead. The mechanism by which the transmitter learns which end of the frontier to sit at is channel feedback, and it is the hidden cost this lesson returns to at the end.

Capacity: The Multiplier, and What It Assumes

For a rich-scattering channel at high SNR, the first-order result is startlingly simple. The capacity of the single channel that M6-L1 gave you is multiplied by the number of independent channels available, and that number is at most the smaller of the two antenna counts — you cannot launch more independent streams than you have transmit antennas, and you cannot resolve more of them than you have receive antennas:

Work it with numbers a learner can hold. Take a 20 MHz channel at 20 dB SNR, which is 10^20/10 = 100 as a power ratio, so the bracket is 1 + 100 = 101. Then log₂(101) = ln(101)/ln(2) = 4.6151/0.6931 = 6.658 bit/s/Hz — check it against the powers of two: 2⁶ = 64 and 2⁷ = 128, so an answer between 6 and 7 is right, and 2^6.658 = 101.0 ✓

• 1×1 (SISO): 20 × 10⁶ × 6.658 = 133.2 Mbps
• 2×2: min(2, 2) = 2, so 2 × 133.2 = 266.3 Mbps
• 4×4: min(4, 4) = 4, so 4 × 133.2 = 532.7 Mbps
• 4×2: min(4, 2) = 2, so 266.3 Mbps — the four transmit antennas buy nothing in rate beyond what two receive antennas can resolve. The smaller count is the one that binds, and it is usually the handset’s

Four antennas at each end, four times the bits, in the same 20 MHz, at the same transmit power, against the same noise floor. That is the reason MIMO is in every modern standard, and it is also the reason MIMO figures are the most over-quoted numbers in wireless — because both of the assumptions behind the multiplier are routinely omitted.

Assumption one: the channel must have rank

The multiplier is at most min(N_t, N_r). What it actually equals is the rank of the channel matrix — the number of genuinely independent equations in it. A pure line-of-sight link with no scattering has rank 1 whatever its dimensions: every receive antenna sees the same wave arriving from the same direction, differing only by a predictable phase, so the rows of the matrix are scaled copies of one another. Feed a 4×4 array a rank-1 channel and you get 133.2 Mbps, not 532.7 — the entire multiplier evaporates, and what remains is the array gain of M7-L4, which is real but is measured in decibels rather than in multiples.

So a MIMO datasheet number assumes a scattering environment your deployment may not have. A 4×4 access point in an open warehouse with a clear view of the client can behave like a 1×1 link with 6 dB of extra gain. The same access point in a partitioned office, where Module 8 said the channel was hostile, will genuinely deliver several streams. Practitioners quote this as rank or rank indication: the receiver measures how many streams the channel can actually support and reports it, and the transmitter obeys. A cell edge with a clean line of sight can therefore be a worse place to be, for throughput, than a cluttered spot nearer in.

Assumption two: the power has to come from somewhere

The formula above holds SNR fixed per stream. If instead you hold the total transmit power fixed and divide it among four streams, each stream loses 10 log₁₀(4) = 6.02 dB. Then the per-stream ratio is 100/4 = 25, the bracket is 26, log₂(26) = 3.2581/0.6931 = 4.700 bit/s/Hz, and the honest total is 4 × 20 × 10⁶ × 4.700 = 376.0 Mbps — not 532.7. That is 376.0/532.7 = 0.706, so 29% below the headline. A larger receive array claws some of it back as array gain, and at low SNR the shortfall is worse still, because log₂(1+x) is nearly linear there and splitting the power gains you almost nothing. Both numbers are computed correctly from the same formula; they differ only in which quantity was held constant, and a specification that does not say which is not a specification.

Diversity: Making the Fades Coincide

The second gain is the one that changes M9-L4’s budget most directly, and its arithmetic is even simpler than the capacity multiplier. A fade only hurts if it hits the path you are using. With N_t × N_r pairs there are that many paths, and if they fade independently then the probability that all of them are deep in a fade at the same instant is the single-branch probability raised to that power:

Take M8-L3’s own figure. A 10 dB fade on a Rayleigh branch has probability p = 1 − e^−0.1 = 1 − 0.9048 = 0.0952, that is 9.52% of the time. Then:

• Diversity order 1 (1×1): 0.0952 = 9.52% — the unprotected case, and the reason M9-L4 carried 8 to 20 dB of fade margin
• Diversity order 2 (2×1 or 1×2): 0.0952² = 0.0952 × 0.0952 = 0.009063 = 0.906%, which is the “roughly 0.9%” M8-L3 quoted for two independent branches ✓
• Diversity order 4 (2×2): 0.0952⁴ = 0.009063² = 8.214 × 10⁻⁵ = 0.0082% — about one moment in 12,000
• Diversity order 16 (4×4): 0.0952¹⁶ = (8.214 × 10⁻⁵)⁴ = 4.6 × 10⁻¹⁷, which is a number no link will ever be limited by

Read the fourth bullet with suspicion rather than delight, because it is where the theory quietly stops describing hardware. Those exponents assume fully independent branches, and sixteen paths through the same room are not sixteen independent random variables. Correlation between antennas reduces the effective diversity order, sometimes drastically; measured 4×4 systems commonly behave like diversity order 6 or 8 rather than 16. The first two rows of that list are trustworthy and are why every base station has had two receive antennas since long before MIMO had a name; the last row is arithmetic, not engineering.

Alamouti: diversity from two transmit antennas

Receive diversity is easy — two antennas, combine what arrives, no cooperation from the far end required. Transmit diversity looks harder, because the two transmitted copies add up in the air and can cancel before the single receive antenna ever sees them. The classic solution is the Alamouti scheme, a space–time block code (STBC) for 2×1. Over two symbol periods it sends a specific pattern — two symbols, then their conjugates with one sign flipped and the antennas swapped — arranged so that a receiver with a single antenna can recover both symbols with full diversity order 2, using only linear processing and needing no channel knowledge at the transmitter at all. That last property is why it shipped: it buys the reliability half of MIMO on the downlink to a one-antenna handset, with no feedback channel. It does not add rate; it sends two symbols in two periods, exactly as a single antenna would.

2×2, 4×4, and the Half-Wavelength Problem

The notation N_t×N_r is universal and its ordering is worth fixing in mind: transmit first, receive second. What ships in real standards is a short list.

Notice that the client is almost always the limit. A 4×4 access point talking to a 2-antenna phone runs at min(4, 2) = 2 streams, so the router box advertising four streams is describing a number the phone in your hand cannot use — unless the access point spends the spare antennas on serving a second device simultaneously, which is exactly what multi-user MIMO does. And the reason phones have two antennas rather than four is physical:

That λ/2 figure is the same number M7-L4 arrived at, and it is important to see that the reason is different. There, half-wavelength spacing was an upper bound: space the elements further apart and grating lobes appear, because the array under-samples the arriving wavefront and direction becomes ambiguous — spatial aliasing. Here, half a wavelength is a rough lower bound for a different purpose: antennas much closer than that see almost the same phasor sum of the same scatterers, their channel coefficients are correlated, and correlated coefficients mean a matrix that is nearly rank-deficient. One constraint is about resolving directions, the other about decorrelating channels, and they happen to land on the same distance because both are set by the wavelength. Practical designs also decorrelate by polarisation rather than by distance, using M7-L2’s orthogonal polarisations to fit two nearly independent channels into one physical location — the standard trick in a cross-polarised base-station panel, and increasingly in handsets too.

Massive MIMO: Many Users, Not Many Streams

M7-L4 ended by noting that a 5G base-station panel carries 64 to 256 elements in a two-dimensional grid, and said that Module 10 was where the story continued. Here it continues. The naive reading of a 64-element panel is “64 streams to one phone”, and that reading is wrong twice over: the phone has two antennas, so min(64, 2) = 2 caps it at two streams, and no single user needs 64 anyway. The actual shift is a change of unit. Massive MIMO stops multiplexing streams to one user and starts multiplexing users onto the same time–frequency resource.

This is multi-user MIMO, and it is spatial multiplexing with the dimension reinterpreted. Sixteen phones scattered around a cell present sixteen different channel vectors to the panel, because they are in different places surrounded by different scatterers. The panel can therefore form sixteen simultaneous beam patterns, each one aimed so that it delivers energy to its own user and lands in a null at the other fifteen. Every user gets the whole channel bandwidth at the same instant. Compare that with M10-L1, where sharing meant dividing frequency, time or codes among users and each user got a fraction: here the resource is divided by space, and each user gets all of it. Space becomes the multiple-access dimension — the fourth one that Module 10 opened with.

Two things make that possible only with a large array. The nulls have to be sharp enough that fifteen users fall into them, which needs many more elements than users — the working rule of thumb is roughly an order of magnitude, hence 64 elements for a handful of simultaneous users. And the panel needs to know all sixteen channel vectors accurately, because a beam pattern is computed from them and a stale or noisy estimate steers the null off the user it was meant to protect, turning a served user into an interferer of the kind M9-L3 catalogued.

That second requirement is where the next lesson becomes relevant. Measuring 64 downlink coefficients per user and feeding them back from the handset is expensive; measuring them once on the uplink and reusing them for the downlink is nearly free, and is legitimate whenever the same frequency is used in both directions, because the propagation channel is reciprocal. Which duplexing arrangement gives you that, and what it costs elsewhere, is M10-L4 — this lesson deliberately stops at naming reciprocity as the thing massive MIMO wants.

The Costs, Honestly

Every gain in this lesson is real and every one is paid for. The bill is why 64-element panels sit on masts and not in pockets.

• One full RF chain per antenna. Each element needs its own power amplifier, its own filters, mixer and data converters. That is the dominant cost, the dominant power draw and the dominant heat load of a massive-MIMO radio, and it does not shrink with clever signal processing. Hybrid architectures reduce it by putting one chain behind a group of elements, at the price of some of the flexibility the full array would have had
• Channel estimation overhead grows with antenna count. Every one of the N_tN_r coefficients has to be measured, which means pilot symbols — known reference signals that carry no user data. The overhead scales with the number of transmit antennas, so a 4×4 link spends roughly four times as much of its capacity on pilots as a 1×1 link. Part of the multiplier you just computed is handed straight back, and the faster the channel changes the more of it goes
• Pilot contamination. There are only so many orthogonal pilot sequences, so in a multi-cell massive-MIMO network they must be reused in neighbouring cells — and then a base station estimating its own user’s channel measures a sum contaminated by a user in the next cell that transmitted the same pilot. The resulting beam is aimed partly at the wrong person. Unlike ordinary noise this does not average away as the array grows, which makes it the characteristic limit of the technique rather than an implementation flaw
• Complexity at the receiver. Unmixing N_t streams means solving a linear system for every symbol, and doing it well means more than that. The processing cost grows quickly with stream count, and it is paid in battery on the device that can least afford it
• The environment is not under your control. Rank is a property of the deployment. A datasheet can promise antennas; it cannot promise scatterers

So treat quoted MIMO figures the way M9-L4 taught you to treat a sensitivity without a bandwidth. Ask which of the three gains is being claimed, how many streams the client supports, what rank the channel actually offers, whether the power was held per-stream or in total, and how much of the frame is pilots. A “4×4, 532 Mbps” badge and a real 250 Mbps in a real room are entirely consistent with each other, and understanding why is the whole point of this lesson.